Matrix Lie groups as 3-dimensional almost contact B-metric manifolds

arXiv:1503.00312v1 [math.DG] 1 Mar 2015

MATRIX LIE GROUPS AS 3-DIMENSIONAL ALMOST CONTACT B-METRIC MANIFOLDS HRISTO MANEV1,2 Dedicated to my doctoral advisor Prof. Dimitar Mekerov on the occasion of his 70th birthday Abstract. The object of investigation are Lie groups considered as almost contact B-metric manifolds of the lowest dimension three. It is established a correspondence of all basic-class-manifolds of the Ganchev-Mihova-Gribachev classification of the studied manifolds and the explicit matrix representation of Lie groups. Some known Lie groups are equiped with almost contact B-metric structure of different types.

Introduction The differential geometry of almost contact manifolds is well studied (e.g. [3]). In [5], the almost contact manifolds with B-metric are introduced and classified. These manifolds can be studied as the odd-dimensional counterpart of the almost complex manifolds with Norden metric [4, 6]. As it is known from [10], every representation of a Lie algebra corresponds uniquely to a representation of the simply connected Lie group. This relation is one-to-one. Then, knowing the representation of a certain Lie algebra settles the question of the representation of its Lie group. In [9], an arbitrary Lie group considered as a manifold is equiped with an almost contact B-metric structure. For the lowest dimension three, it is deduced an equivalence between the classification of considered manifolds given in [5] and the corresponding Lie algebra determined by commutators. In the present work, it is found the explicit correspondence between all basicclass-manifolds of the classification in [5] and Lie groups given by their explicit matrix representation. The paper is organized as follows. In Sect. 1 we recall some preliminary facts about the almost contact B-metric manifolds and Lie algebras related to them. In Sect. 2 we give the correspondence of all basic-class-manifolds and matrix Lie groups. Sect. 3 is devoted to some examples connected to the previous investigations. 1. Preliminaries 1.1. Almost contact B-metric manifolds. Let us denote an almost contact Bmetric manifold by (M, ϕ, ξ, η, g), i.e. M is an (2n + 1)-dimensional differentiable manifold, ϕ is an endomorphism of the tangent bundle, ξ is a Reeb vector field, η 2000 Mathematics Subject Classification. 53C15, 53C50, 53D15. Key words and phrases. Almost contact structure, B-metric, Lie group, Lie algebra, indefinite metric. 1

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is its dual contact 1-form and g is a pseudo-Riemannian metric (called a B-metric) of signature (n + 1, n) such that [5] ϕξ = 0,

ϕ2 = − Id +η ⊗ ξ, η ◦ ϕ = 0, g(ϕx, ϕy) = −g(x, y) + η(x)η(y),

η(ξ) = 1,

where Id is the identity. In the latter equalities and further, x, y, z, w will stand for arbitrary elements of the algebra of the smooth vector fields on M or vectors in the tangent space Tp M of M at an arbitrary point p in M . In [5], it is given a classification of almost contact B-metric manifolds, consisting of eleven basic classes F1 , F2 , . . . , F11 . It is made with respect to the tensor F of type (0,3) defined by
F (x, y, z) = g (∇x ϕ) y, z , where ∇ stands for the Levi-Civita connection of g. The special class F0 , determined by the condition F (x, y, z) = 0, is the intersection of all basic classes and it is known as the class of the cosymplectic B-metric manifolds.
Let {ξ; ei } (i = 1, 2, . . . , 2n) be a basis of Tp M and let g ij be the inverse matrix of (gij ). Then the Lee 1-forms θ, θ∗ , ω associated with F are defined by θ(z) = g ij F (ei , ej , z),

θ∗ (z) = g ij F (ei , ϕej , z),

ω(z) = F (ξ, ξ, z).

In this work, we consider the case of the lowest dimension of the considered manifolds, i.e. dim M = 3. We introduce an almost contact structure (ϕ, ξ, η) on M defined by ϕe1 = e2 , ϕe2 = −e1 , ϕe0 = 0, ξ = e0 , η(e1 ) = η(e2 ) = 0, η(e0 ) = 1 and a B-metric g such that g(e0 , e0 ) = g(e1 , e1 ) = −g(e2 , e2 ) = 1,

g(ei , ej ) = 0, i 6= j ∈ {0, 1, 2}.

The components of F , θ, θ∗ , ω with respect to the ϕ-basis {e0 , e1 , e2 } are denoted by Fijk = F (ei , ej , ek ), θk = θ(ek ), θk∗ = θ∗ (ek ), ωk = ω(ek ). According to [7], we have: θ0 = F110 − F220 , θ0∗ = F120 + F210 , ω0 = 0,

θ1 = F111 − F221 , θ1∗ = F112 + F211 , ω1 = F001 ,

θ2 = F112 − F211 , θ2∗ = F111 + F221 , ω2 = F002 .

MATRIX LIE GROUPS AS 3-DIMENSIONAL ALMOST CONTACT B-METRIC MANIFOLDS 3

If Fs (s = 1, 2, . . . , 11) are the components of F in the corresponding basic classes Fs then: [7]

F1 (x, y, z) = x1 θ1 − x2 θ2 y 1 z 1 + y 2 z 2 , θ1 = F111 = F122 , θ2 = −F211 = −F222 ; F2 (x, y, z) = F3 (x, y, z) = 0; n

F4 (x, y, z) = 12 θ0 x1 y 0 z 1 + y 1 z 0 − x2 y 0 z 2 + y 2 z 0 , 1 2 θ0 = F101 =  F110 = −F202 =
−F220 ;
F5 (x, y, z) = 12 θ0∗ x1 y 0 z 2 + y 2 z 0 + x2 y 0 z 1 + y 1 z 0 , 1 ∗ 2 θ0 = F102 = F120 = F201 = F210 ; (1.1) F6 (x, y, z) = F 7 (x, y, z) = 0;

F8 (x, y, z) = λ x1 y 0 z 1 + y 1 z 0 + x2 y 0 z 2 + y 2 z 0 , λ = F101 =  F110 = F202 = F
220 ;
F9 (x, y, z) = µ x1 y 0 z 2 + y 2 z 0 − x2 y 0 z 1 + y 1 z 0 , µ = F102 = F120 = −F201 =
−F210 ; F10 (x, y, z) = νx0 y 1 z 1 + y 2 z 2
, ν = F011 = F
022 ; F11 (x, y, z) = x0 y 1 z 0 + y 0 z 1 ω1 + y 2 z 0 + y 0 z 2 ω2 , ω1 = F010 = F001 , ω2 = F020 = F002 , where x = xi ei , y = y j ej , z = z k ek . Obviously, the class of three-dimensional almost contact B-metric manifolds is F1 ⊕ F4 ⊕ F5 ⊕ F8 ⊕ F9 ⊕ F10 ⊕ F11 . 1.2. The Lie algebras corresponding to Lie groups with almost contact B-metric structure. Let L and l be a 3-dimensional real connected Lie group and its corresponding Lie algebra. If {E0 , E1 , E2 } is a basis of left invariant vector fields then an almost contact structure (ϕ, ξ, η) and a B-metric g are defined as follows: ϕE0 = 0, ϕE1 = E2 , ϕE2 = −E1 , ξ = E0 , η(E0 ) = 1,

η(E1 ) = η(E2 ) = 0,

g(E0 , E0 ) = g(E1 , E1 ) = −g(E2 , E2 ) = 1, g(E0 , E1 ) = g(E0 , E2 ) = g(E1 , E2 ) = 0.

Then (L, ϕ, ξ, η, g) is a 3-dimensional almost contact B-metric manifold. If its Lie algebra l is determined by k [Ei , Ej ] = Cij Ek ,

i, j, k ∈ {0, 1, 2},

then the components of F , θ, θ∗ , ω are the following: [9] 1 F111 = F122 = 2C12 ,

2 F211 = F222 = 2C12 ,

1 F120 = F102 = −C01 ,

0 F020 = F002 = −C01 ,

2 F210 = F201 = −C02 ,

F110 = F101 = F220 = F202 =

1 2 1 2

0 F010 = F001 = C02 ,
0 2 1 C12 − C01 + C02 ,
0 2 1 , C12 + C01 − C02

0 2 1 F011 = F022 = C12 + C01 + C02 ,

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1 2 θ0∗ = −C01 − C02 ,

2 1 θ0 = −C01 + C02 ,

θ1 =

θ2 =

θ1∗ θ2∗

1 2C12 , 2 −2C12 ,

=

=

ω0 = 0,

2 2C12 , 1 2C12 ,

0 ω1 = C02 , 0 ω2 = −C01 .

Theorem A ([9]). The manifold (L, ϕ, ξ, η, g) belongs to the basic class Fs (s ∈ {1, 4, 5, 8, 9, 10, 11}) if and only if the corresponding Lie algebra l is determined by the following commutators: F1 :

F4 :

F5 :

[E0 , E1 ] = [E0 , E2 ] = 0, [E0 , E1 ] = αE2 ,

[E1 , E2 ] = αE1 + βE2 ;

[E0 , E2 ] = −αE1 ,

[E1 , E2 ] = 0;

[E0 , E1 ] = αE1 ,

[E0 , E2 ] = αE2 ,

[E1 , E2 ] = 0;

F8 :

[E0 , E1 ] = αE2 ,

[E0 , E2 ] = αE1 ,

[E0 , E1 ] = αE1 ,

F10 :

[E0 , E1 ] = αE2 ,

[E0 , E2 ] = −αE2 ,

[E1 , E2 ] = −2αE0 ;

[E0 , E2 ] = αE1 ,

[E1 , E2 ] = 0;

[E0 , E1 ] = αE0 ,

[E0 , E2 ] = βE0 ,

[E1 , E2 ] = 0,

F9 :

F11 :

[E1 , E2 ] = 0;

where α, β are arbitrary real parameters. Moreover, the relations of α and β with the non-zero components Fijk in the different basic classes Fs from (1.1) are the following: F1 : α = 21 θ1 , β = 21 θ2 ; F4 : α = 12 θ0 ; F5 :

α = − 21 θ0∗ ;

F11 :

α = −ω2 ,

F9 :

α = −µ;

F8 :

β = ω1 .

F10 :

α = −λ;

α = 21 ν;

2. The main result Theorem A yields an equivalence between the manifolds of the classification in [5] and the corresponding Lie algebra. It is known (e.g. [10]) that for a real Lie algebra of finite dimension there is a corresponding connected simply connected Lie group, which is determined uniquely up to isomoprhism. It is arised the problem for determination of the Lie group which is isomorphic to the given Lie group L equiped with a structure (ϕ, ξ, η, g) in the class Fs . Theorem 2.1. Let (L, ϕ, ξ, η, g) be an almost contact B-metric manifold belonging to the class Fs (s ∈ {1, 4, 5, 8, 9, 10, 11}). Then the compact simply connected Lie group G isomorphic to L, both with one and the same Lie algebra, has the form eA = E + tA + uA2 , where E is the identity matrix and A is the matrix representation of the corresponding Lie algebra. The matrix form of A as well as the real parameters t and u for the different classes Fs are given in Table 1, where a, b, c ∈ R and α, β are introduced in Theorem A.

2.1. Proof of the theorem.

MATRIX LIE GROUPS AS 3-DIMENSIONAL ALMOST CONTACT B-METRIC MANIFOLDS 5

Table 1. The matrix form of A and the values of t and u for Fs F1 :

F4 :

F5 :

F8 :



 0 0 0 βb  A =  0 αb 0 −αa −βa tr A =  αb − βa  0 −αb αa A= 0 0 −αc  0 αc 0

tr A2 = −2α2 c2   0 αa αb 0  A =  0 −αc 0 0 −αc tr A = −2αc   0 αb αa 0 −αc  A =  −2αb 2αa −αc 0 tr A2 = 2α2 ∆

F9 :

F10

F11

∆ = 2a2 − 2b2 + c2   0 αa −αb A =  0 −αc 0  0 0 αc

tr A2 = 2α2 c2  0 αb 0 : A= 0 0 −αc

tr A2 = 2α2 c2  αa + βb : A =  −αc −βc tr A = αa + βb

 αa −αc  0 0 0 0

 0 0  0

t=



etr A −1 tr A ,

1,

tr A 6= 0 tr A = 0

u = 0 √  sin√ − 12 tr A2 , tr A2 6= 0 − 21 tr A2 t=  1, tr A = 0 √ ( 1−cos − 21 tr A2 , tr A2 6= 0 − 12 tr A2 u= 0, tr A = 0 ( 1 tr A −1 e2 , tr A 6= 0 1 t= 2 tr A 1, tr A = 0 u=0 √  α |∆|  − sin √  , tr A2 < 0  α |∆| t= tr A2 = 0  1, √   sinh√α ∆ , tr A2 > 0 ∆  α √ cos α |∆|−1   , tr A2 < 0 α2 ∆ 1 u= tr A2 = 0 2, √   cosh α ∆−1 , tr A2 > 0 α2 ∆  √ 1 2  sinh √ 1 2 tr2A , tr A2 6= 0 tr A t=  1, 2 tr A2 = 0 ( 2 1 cosh 2 tr A −1 , tr A2 6= 0 1 2 u= 2 tr A 0, tr A2 = 0  sinh αc 2 αc , tr A 6= 0 t= 2 1, tr A = 0  cosh αc 2 α2 c2 , tr A 6= 0 u= 0, tr A2 = 0  − tr A e −1 , tr A 6= 0 tr A t= 1, tr A = 0 u=0

2.1.1. Lie groups as manifolds from the classes F1 , F5 , F11 . Firstly, let us consider the case when (L, ϕ, ξ, η, g) is a F1 -manifold. Then, according to Theorem A, the corresponding Lie algebra is determined by (2.1)

[E0 , E1 ] = [E0 , E2 ] = 0,

[E1 , E2 ] = αE1 + βE2 ,

where α = 21 θ1 , β = 12 θ2 . From (2.1) we have the nonzero values of the commutation coefficients: (2.2)

1 1 C12 = −C21 = α,

2 2 C12 = −C21 = −β.

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According to [10], the commutation coefficients provide a matrix representation of the Lie algebra. This representation is obtained by the basic matrices Mi , which entries are determined by k (Mi )kj = −Cij .

(2.3) Using (2.2) and (2.3), we obtain 

0 M0 =  0 0

 0 0 0 0 , 0 0



0 M1 =  0 0

 0 0 0 0 , −α −β



0 M2 =  0 0

 0 0 α β . 0 0

Let us suppose that (a, b) 6= (0, 0). The matrix representation of the considered Lie algebra g1 is the matrix A, given in Table 1. Then, the characteristic polynomial of A is PA (λ) = λ2 (λ − αb + βa).

Therefore, the eigenvalues λi (i = 1, 2, 3) are λ1 = λ2 = 0,

λ3 = αb − βa.

Hence, the corresponding linearly independent eigenvectors pi (i = 1, 2, 3) are p1 (1, 0, 0)T ,

p2 (0, β, −α)T ,

p3 (0, −b, a)T

and their matrix P has the following form   1 0 0 P =  0 β −b  . 0 −α a

Let us denote ∆ = βa − αb. Then, we have ∆ = det P = − tr A. Let us consider the first case ∆ 6= 0. Then we have   1 0 0 b a P −1 =  0 ∆ ∆  . β α 0 ∆ ∆

The Jordan matrix J is the diagonal matrix with elements Jii = λi . It is known that eA = P eJ P −1 . Then we obtain the matrix Lie group representation G1 of the considered Lie algebra g1 in this case as follows     1 0 0  

βb −∆ −∆  ∆ 6= 0 1 − e 1 − e G1 = eA =  0 1 + αb , ∆ ∆

  βa −∆ −∆ 1 − e 1 − 1 − e 0 − αa ∆ ∆ which can be written as

G1 =



eA = E +

 1 − e−∆ A ∆ 6= 0 . ∆

In the second case ∆ = 0, the matrix P is non-invertible. Then A is nilpotent and eA can be computed directly from eA = E + A +

A3 Aq−1 A2 + + ···+ + , 2! 3! (q − 1)!

MATRIX LIE GROUPS AS 3-DIMENSIONAL ALMOST CONTACT B-METRIC MANIFOLDS 7

where q stands for the degree of A. Then, from the form of A in Table 1 we obtain that q = 2. Therefore, the matrix representation G1 of the Lie group for the Lie algebra g1 in this case is  G1 = eA = E + A, ∆ = 0 , i.e.

 



1 G1 = eA =  0  0

0 1 + αb −αa

  0  βb  , ∆ = 0 .  1 − βa

Both the cases for F1 can be generalized as it is shown in Table 1. By similiar considerations we obtain the results in Table 1 for the classes F5 and F11 . 2.1.2. Lie groups as manifolds from the classes F4 , F9 , F10 . Firstly, let us consider the case when (L, ϕ, ξ, η, g) is a F4 -manifold. Then, according to Theorem A, the corresponding Lie algebra is determined by (2.4)

[E0 , E1 ] = αE2 ,

[E0 , E2 ] = −αE1 ,

[E1 , E2 ] = 0,

where α = 21 θ0 . By similar way as in §2.1.1, we obtain the form of A for the corresponding Lie algebra g4 , given in Table 1. In this class tr A = 0 and we have two cases according to tr A2 = −2α2 c2 . Then, the matrix representation is respectively     1 ac (1 − cos αc) − bc sin αc bc (1 − cos αc) + ac sin αc    , c 6= 0 , G4 = eA =  0 cos αc − sin αc   0 sin αc cos αc     1 −αb αa   1 0 , c = 0 . G4 = eA =  0   0 0 1

Both the cases for F4 can be generalized as it is shown in Table 1. By similiar considerations we obtain the results in Table 1 for the classes F9 and F10 .

2.1.3. Lie groups as manifolds from the class F8 . According to Theorem A, we have [E0 , E1 ] = αE2 ,

[E0 , E2 ] = αE1 ,

[E1 , E2 ] = 0,

where α = −λ. We get the matrix representation of A for the considered Lie algebra g8 , given in Table 1. In this class tr A = 0 and we have three cases according to the sign of tr A2 = 2 2α ∆, where ∆ = 2a2 − 2b2 + c2 . They can be generalized as it is shown in Table 1. 3. Equipping of known Lie groups with almost contact B-metric structure In this final section we equip some known Lie groups with almost contact Bmetric structures of different types. These considerations use results given in [8].

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3.1. Examples 1, 2, 3. In [11], there are considered the matrix Lie GII and GIII of the following form  −z     e 0 x cos z − sin z x 1 ez y  , GII =  sin z cos z y  , GIII =  0 GI =  0 0 0 1 0 0 1 0

groups GI ,  x y 1 z , 0 1

where x, y, z ∈ R. These matrix groups and their automorphisms represent the only three types of generalized affine symmetric spaces denoted as I, II, III. The Lie group GI is the group of hyperbolic motions of the plane R2 , GII is the group of isometries of the plane R2 and GIII is the Heisenberg group. The corresponding Lie algebras are determined by the commutators as follows: gI : gII : gIII :

[X1 , X3 ] = X1 , [X1 , X3 ] = −X2 , [X1 , X3 ] = X2 ,

[X2 , X3 ] = −X2 , [X2 , X3 ] = X1 , [X2 , X3 ] = 0,

[X1 , X2 ] = 0, [X1 , X2 ] = 0, [X1 , X2 ] = 0.

The types of the Lie algebras gI , gII , gIII according to the well-known Bianchi classification in [1, 2] of three-dimensional real Lie algebras are Bia(V), Bia(VII0 ), Bia(II), respectively. Bearing in mind Table 1 for F9 and substituting X1 = E1 , X2 = E2 , X3 = −E0 , α = 1, we have that GI can be considered as an almost contact B-metric manifold of the class F9 when the derived algebra is on ker(η), otherwise the manifold belongs to F1 ⊕ F11 . [8] Considering Table 1 for F4 and substituting X1 = E1 , X2 = E2 , X3 = E0 , α = 1, we have that GII can be considered as an almost contact B-metric manifold of the class F4 when the derived algebra is on ker(η), otherwise the manifold belongs to F4 ⊕ F8 or F4 ⊕ F8 ⊕ F10 . [8] Depending on the way of equipping with an almost contact B-metric structure we can obtain GIII as a manifold in F4 ⊕ F10 or F8 ⊕ F10 when the derived algebra is on ker(η) or span(ξ), respectively. [8] 3.2. Example 4. Let us consider the well-known Lie group SO(3) which is called the rotation group. The matrix representation A of the corresponding Lie algebra has the form   0 −αz αy 0 −αx  , A =  αz −αy αx 0 where α is the angle of rotation and x, y, z ∈ R. In terms of the matrix exponential and according to the Rodrigues rotational formula, the Lie group SO(3) consists the matrices of the following form eA = E + sin α A + (1 − cos α) A2 .

Then, the type of the considered Lie algebra is Bia(IX), according to the Bianchi classification. Hence, bearing in mind [8], this Lie group can be considered as an almost contact B-metric manifold of the class F4 ⊕ F8 ⊕ F10 . References [1] Bianchi, L., Sugli spazi a tre dimensioni che ammettono un gruppo continuo di movimenti, Memorie di Matematica e di Fisica della Societa Italiana delle Scienze, Serie Terza, 11 (1898), 267-352. [2] Bianchi, L., On the three-dimensional spaces which admit a continuous group of motions, Gen. Rel. Grav., 33 (2001), 2171-2253.

MATRIX LIE GROUPS AS 3-DIMENSIONAL ALMOST CONTACT B-METRIC MANIFOLDS 9

[3] Blair, D.E., Riemannian Geometry of Contact and Symplectic Manifolds, Progress in Mathematics, Birkhauser, Boston, 2002. [4] Ganchev, G., Borisov, A., Note on the almost complex manifolds with a Norden metric, C. R. Acad. Bulg. Sci., 39 (1986), 31-34. [5] Ganchev, G., Mihova, V. and Gribachev, K., Almost contact manifolds with B-metric, Math. Balkanica (N.S.), 7 (1993) no. 3-4, 261-276. [6] Gribachev, K., Mekerov, D., Djelepov, G., Generalized B-manifolds, C. R. Acad. Bulg. Sci., 38 (1985), 299-302. (1985) [7] Manev, H., On the structure tensors of almost contact B-metric manifolds, arXiv:1405.3088. [8] Manev, H., Almost contact B-metric structures and the Bianchi classification of the threedimensional Lie algebras, arXiv:1412.8142. [9] Manev, H. and Mekerov, D., Lie groups as 3-dimensional almost contact B-metric manifolds, J. Geom., Online September 2014, DOI:10.1007/s00022-014-0244-0. [10] Gilmore, R., Lie groups, Lie algebras and some of their applications, A Wiley-Interscience Publication, New York, 1974. [11] Kowalski, O., Generalized Symmetric Spaces, Lecture Notes in Mathematics, 805 (Eds: A. Dold, B. Eckmann), Springer, Heidelberg, 1980. (1) Medical University of Plovdiv, Faculty of Pharmacy, Department of Pharmaceutical Sciences, 15-A Vasil Aprilov Blvd., Plovdiv 4002, Bulgaria; (2) Paisii Hilendarski University of Plovdiv, Faculty of Mathematics and Informatics, Department of Algebra and Geometry, 236 Bulgaria Blvd., Plovdiv 4027, Bulgaria E-mail address: [email protected]